Computing coefficients

Binomial[-s, 0]

Integrate[Binomial[-s, 2], {s, 0, 1}]

Table[((-1)^k ) 
        Integrate[Binomial[-s, k], {s, 0, 1}], {k, 0, 10}]

Four-step Adam-Bashforth

First, test the code to initialize the Adam-Bashforth method by computing three steps using Midpoint.

f[t_, y_] :=  10  y (1 - y);
t0 = 0. ;
y0 = 0.5;
tf = 2;
n = 20;
h = (tf - t0)/n;
tylist = {{t0, y0}};
Do[
tylist = 
          Append[tylist, 
{t + 
                h, 
tylist[[-1, 2]] + 
                h*f[t + 0.5*h, 
                    tylist[[-1, 2]] + 
                      0.5*h*f[t, tylist[[-1, 2]]]]}], {t, t0, 
          t0 + 2  h, h}];

tylist

Expand the previous code to do Adam-Bashforth for remaining steps.

f[t_, y_] :=  10  y (1 - y);
t0 = 0. ;
y0 = 0.1;
tf = 1;
n = 20;
h = (tf - t0)/n;
tylist = {{t0, y0}};
  
Do[
tylist = 
          Append[tylist, 
{t + 
                h, 
tylist[[-1, 2]] + 
                h*f[t + 0.5*h, 
                    tylist[[-1, 2]] + 
                      0.5*h*f[t, 
                          tylist[[-1, 
                              2]]]]
}], 
{t, t0, t0 + 2  h, h}];
 (* 
      Adam - Bashforth 4 - 
        step method for remaining*) 
Do[
tylist = 
          Append[tylist, 
{t + 
                h, 
tylist[[-1, 2]] + 
                h/24*(55*f[t, tylist[[-1, 2]]] - 
                      59*f[t - h, tylist[[-2, 2]]] + 
                      37*f[t - 2  h, tylist[[-3, 2]]] - 
                      9*f[t - 3  h, 
                          tylist[[-4, 
                              2]]])
}], 
{t, t0 + 3  h, tf - h, h}];

tylist // TableForm

approxsolnplot = 
      ListPlot[tylist, PlotStyle -> RGBColor[1, 0, 0], 
        PlotJoined -> True]